Plasma Spectroscopy-Takashi Fujimoto

If we are concerned with the intensitydistribution of spectral lines over a spectrum, we have to understand the popu-lation density distribution over the excited levels of atoms and ions

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Plasma Spectroscopy

TAKASHI FUJIMOTO

Department of Engineering Physics and Mechanics

Graduate School of Engineering

Kyoto University

CLARENDON PRESS O X F O R D

2004

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Throughout the history of spectroscopy, plasmas have been the source of tion, and they were studied for the purpose of spectrochemical analysis and alsofor the investigation of the structure of atoms (molecules) and ions constitutingthese plasmas About a century ago, the spectroscopic investigations of theradiation emitted from plasmas contributed to establishing quantum mechanics.However, the plasma itself has been the subject of spectroscopy to a lesser extent

radia-This less-developed state of plasma spectroscopy is attributed partly to the

com-plicated relationships between the state of the plasma and the spectral acteristics of the radiation it emits If we are concerned with the intensitydistribution of spectral lines over a spectrum, we have to understand the popu-lation density distribution over the excited levels of atoms and ions in the plasma.Since the latter distribution is governed by a collection of an enormous number ofatomic processes, e.g electron impact excitation, deexcitation, ionization,recombination, and radiative transitions, and since the spatial transport of theplasma particles and the temporal development are sometimes essential as well, it

char-is rather difficult, by starting from these elementary processes, to deduce astraightforward consequence concerning the population distribution For certainlimiting conditions of the plasma, e.g for the low- or high-density limit, several

concepts like corona equilibrium and local thermodynamic equilibrium have been

proposed, but they have been accepted on a rather intuitive basis

This book is intended first to provide a theoretical framework in which we cantreat various features of the population density distribution over the excited levels(and the ground state) of atoms and ions and give their interpretation in a unifiedand coherent way In this new framework several concepts, some of which arealready known and some newly derived, are properly defined For these purposes,

we take hydrogen-like ions (and neutral hydrogen) as an example of an ensemble

of atoms and ions immersed in a plasma Following the first three introductorychapters, these problems are discussed in the subsequent two chapters The fol-lowing three chapters are devoted to several facets which are useful in performing

a spectroscopy experiment This volume concludes with a chapter treating severalphenomena characteristic of dense plasmas This chapter may be regarded as anapplication of the theoretical methods developed in the first part of the volume.The main body of this book is based on my half-year course given at theGraduate School, Kyoto University, for more than a decade This book isintended mainly for graduate students, but it should also be useful for researchersworking in this field A reader who wants to obtain only the basic ideas may skipthe chapters and sections marked with an asterisk

In writing this book I owe thanks to many colleagues and students First of all,Professor Otsuka is especially thanked for his careful reading of the wholemanuscript and for pointing out errors, and giving me critical comments andvaluable suggestions Professor Kato and M Goto provided me with their valu-able unpublished spectra for Chapter 1 Various materials in Chapters 4 and 5were taken from publications by my former students, K Sawada, T Kawachi, and

M Goto These and A Iwamae, my present colleague, created many beautifulfigures for this book I am also grateful to Dr Baronova for her comments, whichhave made this book more or less comprehensive She also helped in some parts ofthe book If this book is quite straightforward for beginners, that is due to mystudents, Y Kimura and M Matsumoto, who gave me various comments andquestions as students I would like to express my thanks to workers who permitted

me to reproduce their figures in this book Professors Xu and Zhu even modifiedtheir original figure so as to fit better into the context of this book Mrs Hooper Jr.who gave me permission to use a figure on behalf of her recently deceased hus-band The names of these workers and the copyright owners are mentioned in thereference section and the figure captions in each chapter

*3.5 Autoionization, dielectronic recombination, and satellite lines 64

*3.6 Ion collisions 72Appendix 3A Scaling properties of ions in isoelectronic sequence 76

*Appendix 3B Three-body recombination "cross-section" 79

4 Population distribution and population kinetics 83

4.1 Collisional-radiative (CR) model 834.2 Ionizing plasma component 964.3 Recombining plasma component - high-temperature case 1114.4 Recombining plasma component - low-temperature case 1204.5 Summary and concluding remarks 131

*Appendix 4A Validity of the statistical populations among

the different angular momentum states 134

*Appendix 4B Temporal development of excited-level populations

and validity condition of the quasi-steady-state approximation 136

5 Ionization and recombination of plasma 150

5.1 Collisional-radiative ionization 1515.2 Collisional-radiative recombination - high-temperature case 1575.3 Collisional-radiative recombination - low-temperature case 163

5.4 lonization balance 1675.5 Experimental illustration of transition from ionizing

plasma to recombining plasma 182Appendix 5A Establishment of the collisional-radiative

rate coefficients 188Appendix 5B Scaling law 190

*Appendix 5C Conditions for establishing local thermodynamic

equilibrium 191

*Appendix 5D Optimum temperature, emission maximum,

and flux maximum 202

6 Continuum radiation 205

6.1 Recombination continuum 2056.2 Continuation to series lines 2076.3 Free-free continuum - Bremsstrahlung 211

*7 Broadening of spectral lines 213

7.1 Quasi-static perturbation 2147.2 Natural broadening 2187.3 Temporal perturbation - impact broadening 2197.4 Examples 2247.5 Voigt profile 233

*8 Radiation transport 236

8.1 Total absorption 2368.2 Collision-dominated plasma 2408.3 Radiation trapping 245Appendix 8A Interpretation of Figure 1.5 252

*9 Dense plasma 257

9.1 Modifications of atomic potential and level energy 2579.2 Transition probability and collision cross-section 2619.3 Multistep processes involving doubly excited states 2669.4 Density of states and Saha equilibrium 277

Index 286

LIST OF SYMBOLS AND ABBREVIATIONS

a 0 first Bohr radius

au atomic units

A a (p,nl 1 ) autoionization probability for (p,nl')

A(p, q) Einstein's A coefficient or transition probability for p —> q

A L line absorption

A r stabilizing radiative transition probability

B(p, q) Einstein's B coefficient for photoabsorption and for induced

E kinetic energy of an electron, energy of level

E G energy of Griem's boundary level with respect to the ground state

E(p, q) energy separation between level p and q

Ei(–x) exponential integral

f(u), f(E) electron velocity (energy) distribution function

f q oqscillator strength for transition p —> q

f p,c oscillator strength for photoionization from level p

F hctric field strength of the plasma microfield

F 0 normal field strength

F(q,p) deexcitation rate coefficient

G scale factor for excitation or deexcitation rate coefficient

g(p) statistical weight of level p

g(E/R) density of states per unit energy interval

ge degeneracy of electron (=2)

gbb, gbt, gft Gaunt factor

G(a) reduced density of states

h Planck's constant, ratio of quasi-static broadening to impact

broadening

h Planck's constant divided by 2p

H scale factor for radiative decay rate

/ scale factor for continuum radiation

n 0 (p) recombining plasma component

n1(p) ionizing plasma component

2 parabolic quantum numbers

N perturber particle density, density of ground-state atoms

p designation of a level, momentum of an electron

PG Griem's boundary level

/IB Byron's boundary level

P Rp (v) recombination continuum radiation power

P Lp (v) line radiation power

P R+B (v) radiation power of recombination continuum and Bremsstrahlung

r d(p, nl') dielectronic capture rate coefficient for (p, nl')

SCR collisional-radiative ionization rate coefficient

S(p) ionization rate coefficient

t res response time of populations of excited levels

t r1 (p) relaxation time of population n(p)

ttr(p) transient time for population of level p

T(x, y) Stark profile with ion and electron broadening

u excitation or ionization energy in threshold units

v speed of an electron

W three-body recombination flux, equivalent width

W(b) field distribution function

z ze is the nuclear charge of the next ionization-stage ion

Z Ze is the charge of perturber particles

Z(p) Saha-Boltzmann coefficient

Z p (pq) Saha-Boltzmann coefficient with respect to the energy position of

level p

a fine structure constant

aCR collisional-radiative recombination rate coefficient

a(p) three-body recombination rate coefficient

density of ions in the next ionization stage

population (density) of level p

LIST OF SYMBOLS AND ABBREVIATIONS xi

reduced electron temperature, T e /z 2

correction factor to Saha equationimpact parameter

impact parameter of Weisskopf radiusmean distance between perturbersreal and imaginary parts of the impact broadening cross-sectionexcitation or deexcitation cross-section

photoabsorption cross-section for excitation p —> q

photoionization cross-sectionradiative recombination cross-sectionionization cross-section

mean free time

period of one revolution of the electron in level n

period of one revolution of the electron in the first Bohr orbittransit time

optical thicknessatomic unit velocityautocorrelation functionintensity or radiant flux or radiant power of emission radiation

for transition p —> q ionization potential of level p

central (angular) frequency of a spectral linecollision strength

free electronindicating the quantity for neutral hydrogenquantity in the low-density limit

quantity in region IIquantity in region IIIByron's boundaryDoppler

relationship valid in thermodynamic equilibriumh

suffixG Griem's boundary

DL dielectronic capture ladder-like

FWHM full width at half-maximum

l.h.s left-hand side

LTE local thermodynamic equilibrium

QSS quasi-steady state

r.h.s right-hand side

INTRODUCTION

1.1 Historical background and outline of the book

The history of spectroscopy began more than three hundred years ago with theexperiment by Newton in which sunlight was dispersed by a prism into light rayswhich bore the colors of a rainbow (Fig 1.1) Later, mainly in the nineteenthcentury, when the instrument called the spectroscope was used to observe thespectra of radiation emitted from various plasmas, i.e flames, the Sun and severalstars, and later electric arcs and sparks (see Fig 1.2), an enormous number ofspectral lines were found as emission or absorption lines As a result of the invention

of the photographic plate, or of the spectrograph, spectroscopy developed into ascience of very high precision in terms of wavelength of observed lines Numerousattempts were made to find regularities manifested by these lines In the beginning

of the twentieth century the experimentally established laws governing thewavelengths, or the frequencies, of the lines characteristic of atoms and ions,together with the spectral characteristics of the black-body radiation, played anessential role in establishing quantum mechanics

Atomic spectroscopy, which deals primarily with wavelengths of spectral lines,

is still actively studied to establish the energy-level structure of complicated atoms

FIG 1.1 The sketch of "the critical experiment" drawn by Newton himself.(By permission of the Warden and Fellows, New College, Oxford.)

FIG 1.2 The "map" of various plasmas NFR means the future nuclear fusion

plasma; "laser" means laser-produced plasmas The oblique line shows thescaling law for neutral hydrogen and hydrogen-like ions according to thenuclear charge z See the text for details

and highly ionized ions The intensities of these lines are of concern mainly fromthe viewpoint of determining the ionization stage of the ions emitting the line andthe strength of the transition, i.e the oscillator strength and the multipolarity.(These terms are explained in a subsequent part of this book.) Other character-istics of the spectrum, e.g broadening of the lines, have less significance to atomicspectroscopists

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK 3The reason why characteristics except for the wavelengths are unimportant inatomic spectroscopy lies in the fact that these spectral characteristics are ephem-eral rather than basic, or they are dependent on the conditions under which thisparticular plasma is produced and on the parameter values this plasma has This

very fact constitutes the starting point of plasma spectroscopy Thus, plasma

spec-troscopy deals with these variable characteristics of the radiation emitted fromthe plasma in relation to the plasma itself, which is regarded as an environment ofthe atoms and ions emitting the radiation

It may be interesting to note that the intensity (the radiant flux or the radiantpower is a more precise term; see later) has been a quantity which is difficult todetermine experimentally In particular, its absolute value could be determinedonly in favorable situations However, developments in techniques, i.e photo-multipliers for the last half-century, and multichannel detectors with digital signalprocessing techniques in recent years, have enabled us to perform quantitativespectroscopy much more easily This situation is favorable for the development ofquantitative plasma spectroscopy as treated in this book

When we look at a plasma, laboratory or celestial as shown in Fig 1.2, through

a spectrometer, we find a spectrum of radiation emitted from this plasma This is a

pattern of spectral lines (and continuum), with varying intensities, distributed over

a certain wavelength range Figure 1.3 shows an example of the spectrum from aplasma This plasma is produced from a helium arc discharge plasma streamingalong magnetic field lines into a dilute helium gas The distribution pattern of lines

in terms of wavelength reflects, of course, the energy-level structure of atoms, orthe composition of the plasma: what atomic species constitute the plasma Thespectrum of Fig 1.3 is of neutral helium Figure 1.4 is the energy-level diagram,called the Grotorian diagram, of neutral helium It is straightforward to identifythe lines in Fig 1.3 with transitions each connecting two levels in this diagram.The thin solid lines show these identifications

We sometimes find that two plasmas having identical wavelength-distributionpatterns show different intensity-distribution patterns Figure 1.5 shows an example;both the spectra are of the resonance series lines of ionized helium (hydrogen-likehelium), terminating on the ground state as shown in Fig 1.6 The plasmasproducing these spectra are essentially the same as that for Fig 1.3 It may be saidthat the two plasmas in Fig 1.5 are of the same composition but show different

"colors" This difference might be attributed to different temperatures of theseplasmas; a plasma with a higher temperature tends to emit intense lines havingshorter wavelengths This suggestion may be supported for two reasons: first, thehigher the energy of electrons in the plasma the higher the energy of atomic statesexcited by them (see Fig 1.6) and thus the shorter the wavelengths of the lightoriginating from these states; second, the higher the temperature the shorter thepeak wavelength of the black-body radiation, i.e Wien's displacement law (seeChapter 2) We will see later (Chapters 5 and 8) whether our conclusion here isadequate or not

emitted from a helium plasma Several series of lines of neutral helium and recombination continua are seen (Plasma

produced in the TPD-I machine, Institute for Plasma Physics, Nagoya Quoted from Otsuka M., 1980 Japanese Journal of

Optics (in Japanese), 9, 149; with permission from The Japanese Journal of Optics.)

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK 5

FIG 1.4 Energy-level diagram of neutral helium The thin solid lines show thetransitions corresponding to the emission lines in Fig 1.3 The dashed linesshow the transitions of Fig 3.6 The dotted line shows the emission line whichappears in Fig 5.17(b) The dash-dot lines appear as emission lines in Figs 7.4and 7.6

When we look closely at a single spectral line, we sometimes find that itsintensity is distributed over a narrow but finite wavelength region, with its peakdisplaced from the original position of the line where it is found under normalconditions We can find several broadened lines in Fig 1.3 in the longer-wavelength region, as contrasted with the accompanying sharp lines We also findprominent examples in Fig 1.7

We may also find continuum spectra underlying the spectral lines An example

is seen in Fig 1.3 Figure 1.7 shows another example For certain plasmas likethese examples the continuum is weaker than the lines but for some others it iseven stronger than the lines

The plasma of Fig 1.7 is produced from a hydrogen pellet (frozen hydrogenice) injected into a high-temperature plasma A dense plasma is produced fromevaporated hydrogen Several broadened lines, tending to a continuum, of neutralhydrogen atoms are seen These lines correspond in Fig 1.6 to the transitions

FIG 1.5 Two spectra from plasmas produced under slightly different conditions.The spectral lines are the resonance series lines (1 2S —«2P) of ionized helium.(TPD-I By courtesy of Professor T Kato.) The asterisk shows the real peakposition when the saturation effect of the detector is corrected for.

terminating on the n = 2 levels Of course, the transition energies are about

one-quarter for these lines, because Fig 1.6 is for hydrogen-like helium ions, notneutral hydrogen in Fig 1.7

As mentioned earlier, all these variable characteristics of the spectrum ofatoms and ions are dependent on the nature of the plasma which emits theradiation In other words, the spectrum contains information about the plasma:i.e it is the fingerprint of the plasma This notion constitutes the basis of plasmadiagnostics or using the observed spectrum to infer the characteristics of theplasma, e.g its temperature, density, and particle transport property over space

The first task of plasma spectroscopy would naturally be to find and establish

the relationships between the characteristics of the emission-line (and continuum)intensities from a plasma and the nature of this plasma Since a spectral line

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK

FIG 1.6 Energy-level diagram of ionized helium, where the different / and mi states are reduced to a single level specified by n The series of transitions

terminating on the ground state correspond to the emission lines in Fig 1.5

The transitions terminating on the n = 1 level correspond to the emission lines

in Fig 1.7, except that this diagram is for hydrogen-like helium and Fig 1.7 isfor neutral hydrogen

is emitted by excited atoms or ions (see Figs 1.4 and 1.6) its intensity is given bythe number density of these atoms or ions Here we have assumed that theidentification of the spectral line, or the correspondence of the line to the upperand lower levels of atoms or ions, is established and that the transition probability

is known for this transition We have also ignored intricate factors, like ization and reabsorption of radiation (see Chapter 8), both of which can affect theobserved line intensity Thus, the problem of the intensity distribution reduces to

polar-that of the population-density (called simply the population henceforth)

distribu-tion of atoms and ions over excited levels (and in the ground state, too) Oncethese relationships are established, several conventional concepts will turn out to

be incorrect For example, the interpretation mentioned above concerning thetemperature and the "color" (Fig 1.5) will be found to be too naive; sometimes ahot plasma may look more "red" than some colder plasmas having the samecomposition

7

FIG 1.7 A spectrum of neutral hydrogen atoms from a plasma produced bypellet (a solid hydrogen ice) injection into a high-temperature plasma.(Produced at the LHD in the National Institute for Fusion Science, Toki Bycourtesy of Dr M Goto.)

The primary objective of this book is to provide the reader with a sound basisfor interpreting various features manifested by a spectrum of radiation emittedfrom a plasma in terms of the characteristics of the plasma

The first two chapters are intended so that the reader acquires the backgroundnecessary to proceed to the main part of the book developed in subsequentchapters First, thermodynamic equilibrium relationships are discussed for thediscrete-level populations, for the ionization balance, and for the radiation field.The subsequent chapter discusses the atomic processes important in plasmas,i.e the spontaneous radiative transition and the transitions due to electron impact

It is pointed out that, for a pair of levels, a single parameter, the absorptionoscillator strength, which gives the radiative transition probability, also determinesthe collisional excitation cross-section, although to a limited extent Anotherimportant fact worth noting is that various features associated with high-lyingexcited states continue smoothly across the ionization limit to those associatedwith low-energy continuum states This is a natural consequence of the continuity

of the corresponding wavefunctions of the atomic electron and the ion (thecontinuum-state electron)

Chapters 4 and 5 present a theoretical framework in which the experimentallyobserved population distribution is interpreted in terms of various characteristics

of the plasma In Chapter 4 we introduce the method known as the radiative (CR) model or the method of the quasi-steady-state (QSS) solution

collisional-HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK 9

By this method we treat in a coherent manner the population couplings among theexcited levels (and the ground state) in the population formation by the collection

of atomic processes Figure 1.8 shows schematically the energy-level structure; p

or q denotes a level and/> = 1 means the ground state As an example of ensembles

of atoms and ions immersed in a plasma and emitting radiation we take like ions (and neutral hydrogen) for the purpose of illustration A discussion ofthe validity of this method, or of the QSS solution, is given in Appendix 4B Then

hydrogen-an excited-level population is expressed as a sum of the ionizing plasma nent and the recombining plasma component Figure 1.9 shows schematically thestructure of the populations For both the components the population distributionand its kinetics are examined in detail Figure 1.10 is the "map" of the populations

compo-of these components, or the summary compo-of our investigations in Chapter 4 Severalcharacteristic population distributions, e.g the minus sixth power distribution, andthe corresponding population kinetics, i.e the ladder-like excitation-ionization

FIG 1.8 Schematic energy-level diagram of an atom or ion with symbols used inthis book

FIG 1.9 The structure of the excited-level populations in the collisional-radiative

model The population n(p) is the sum of the ionizing plasma component n^(p)

which is proportional to the ground-state population «(1) and the recombining

plasma component n 0 (p) proportional to the "ion" density n z Full

explana-tions are given in Chapter 4

the summary of our investigations to be developed in Chapter 4, so that a reader who has started to read this book doesnot need to understand the details of this diagram The abscissa is the (reduced) electron density and the ordinate is theprincipal quantum number of excited levels, (a) The ionizing plasma component; (b) the recombining plasma component.

Griem's boundary p G , given by eq (4.25), (4.29), or (4.59), divides the whole area into a low-density region and

high-density region Byron's boundary />B, given by eq (4.55) or (4.56), divides the high-density region into low-lying levels andhigh-lying levels In each area, the name of the phase I, the population distribution, and the dominant population kinetics

are shown for level p with which we are concerned For the capture-radiative-cascade (CRC) phase in (b) the

near-Saha-Boltzmann population is for the high-temperature case In practical situations of the ionizing plasma (a) Byron's

boundary lies far below p = 2, and only the saturation phase with the multistep ladder-like excitation-ionization mechanism appears in the high-density region (Quoted from Fujimoto, T 1980 Journal of the Physical Society of Japan,

49, 1591, with permission from The Physical Society of Japan.)

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK 11mechanism, are established Two important boundaries, Griem's boundary andByron's boundary, are derived for electron density and temperature, or for excitedlevels It is noted that the strong population coupling among the excited levels andits continuation to the ionic (continuum) states play an essential role in deter-mining the population distributions in both the components.

In a plasma an ensemble of atoms or ions as a whole may be in a dynamicalprocess of ionization or recombination, depending on the time history and thespatial structure of the plasma In these processes, in addition to the direct ioniza-tion and recombination, excited levels play essential roles, too, and they affect theeffective rate of ionization and that of recombination This subject is examined inChapter 5 An important finding is that the ionization process is associated withthe ionizing plasma component of populations and thus the magnitude of theionization flux is proportional to the excited-level population A similar propor-tionality is also valid for the recombination flux and the recombining plasmacomponent In both the cases, the proportionality factors naturally depend on theparameters of the plasma Thus, an emission line intensity is a measure of theionization flux and that of the recombination flux, depending on the nature ofthe plasma

There is a class of plasmas in which the ionization flux and the recombinationflux balance with each other, or plasmas in ionization balance An importantconclusion is reached for this class of plasmas: the ratio of the contributions to anexcited-level population from the ionizing plasma component and from therecombining plasma component could be comparable in magnitude This findingleads to another even more important conclusion: for many actual plasmas whichare out of, sometimes far from, ionization balance, only one of the two compo-nents dominates the actual populations while the other gives a negligibly smallcontribution Thus we have reached an important step for a correct explanation ofthe different spectra in Fig 1.5 Another important finding is that, among plasmas

in various states of ionization-recombination, a plasma in ionization balance givesrise to the minimum of radiation intensity Plasmas out of ionization balance emitmuch stronger radiation

Following these two chapters of primary importance, we now turn to otherfacets which together constitute plasma spectroscopy Chapter 6 treats the con-tinuum radiation The spectral characteristics are examined for the recombinationcontinuum, and a smooth continuation is established for its intensity to those ofthe accompanying series lines, as is observed in Figs 1.3 and 1.7 This con-tinuation is interpreted as the continuation of the "populations" of the discretestates to the continuum states

The problems of broadening and shift of spectral lines follow in Chapter 7

We have already seen some examples in Figs 1.3 and 1.7 This aspect is importantfor determining the atom (ion) temperature or the plasma density Besides theDoppler broadening and Stark broadening in the quasi-static approximation,natural broadening and impact broadening is treated in a rather elementary way.The latter class of broadening is regarded as the relaxation of optical coherence

Chapter 8 treats the phenomena associated with radiation transport We firstexamine how the absorption line profile develops in an absorbing medium.Then we describe how the observed intensity and profile of an emission linedevelops in a plasma when the plasma becomes optically thick to this line We thenconsider the situation in which the excited-level population is controlled by asequence of processes of emission and reabsorption of radiation, i.e radiationtrapping We examine the phenomenon on the basis of two approaches whichare complementary to each other At this point we are able to give the correctinterpretation to the spectra in Figs 1.5 and 1.7.

When our plasma is dense, various new phenomena may appear which areabsent in "everyday" plasmas Although part of this problem has already beendiscussed in the context of excited-level populations and line broadening, furtherdiscussions deserve a separate chapter In Chapter 9 we investigate how the atomicstate energy and the collision cross-section are affected by the screening of theCoulomb interactions by the plasma particles surrounding the atom or the ion

We then examine additional new excitation and deexcitation processes of ionsinvolving the doubly excited states Contributions from the resonance process tothe excitation cross-section are also found to be affected in a dense plasma Directrecombination of ions in an excited level can be important under certain condi-tions Finally, we investigate modification to the density of atomic states overenergy and its consequence incurred in the thermodynamic equilibrium relation-ship of the densities of atoms and ions in the plasma, a modification to the result

of Chapter 2

1.2 Various plasmas

Figure 1.2 shows various kinds of plasmas on the plane n e —T e , Its abscissa and

ordinate are the most important parameters of a plasma: the number density ofelectrons, or simply the electron density, «e, in units of m~3, or in cm~3, and the

electron temperature, T e , in units of K Occasionally, kT e expressed as eV tron volts: the energy of an elementary charge e accelerated by a potential of 1 V)

(elec-is used: kT e = 1 eV corresponds to T e = 11,605 K The abscissas and the ordinates

of Fig 1.2 are expressed in these units So, a plasma is located somewhere on this

plane according to its n e —T e values The most modest plasmas are flames likecandles, and those in internal combustion engines, which are produced and heated

by chemical reactions We have enormous numbers of plasmas produced byelectric discharge The class of glow discharges, which are produced in a low-pressure gas, includes many laboratory plasmas as well as plasmas encountered inour everyday life; an example is the plasma in fluorescent lamp tubes According

to the discharge current drawn, «e varies over several orders, but T e lies in a rather

narrow range, and kT e is one through a couple of eV Later in Chapter 4, we willencounter an example of this class of plasma Many kinds of processing plasmas,which are used for the purpose of manufacturing, e.g semiconductor devices, areproduced by radio-frequency or microwave discharges in chemically active gases

NOMENCLATURE AND BASIC CONSTANTS 13They have similar parameter values If an electric discharge is made continuously

in a gas under atmospheric or even higher pressure, we have arc discharge plasmas.This kind includes many kinds of lamps for illumination, e.g mercury discharge

lamps Interestingly, T e of these plasmas has a very narrow range around

kT e = 1 eV The plasma shown in Fig 1.7 happens to be very similar to this class.

When the heating of electrons by electric power input stops, the plasma decays intime or, in the case of a flowing plasma, in space The plasmas in these decaying

processes are called afterglows, and have low T e In Chapter 4, we will find a few

examples of this class of plasma It will turn out that the plasma of Fig 1.3 alsobelongs to this class If an electric breakdown of a high voltage takes place in anatmospheric-pressure gas, we have a spark discharge, or even lightning Owing tothe sudden input of energy into a thin area of the gas, a plasma with a rather high

Te is produced

In contrast to these "classical" plasmas we now have more "powerful" plasmaswhich have been developed in the last couple of decades One of the motivations ofthese developments came from the possibility of realizing nuclear fusion reactionsfor a future energy source One class of such plasmas is called the magneticallyconfined plasma A high-temperature and moderate-density plasma is confinedwithin the toroidal-shaped vessel made with a magnetic field With the enormousprogress in scientific and technological developments, plasma machines with theconfiguration called tokamak now produce plasmas close to the practical conditionfor nuclear fusion reactions Helical configuration plasmas are also being vigor-ously investigated In Fig 1.2 the NFR region means the parameters of the futurenuclear fusion reactor Another class is high-energy-density plasmas which areproduced by putting a vast amount of energy into a small volume of a gas or asolid in a very short time A rather traditional approach is pinch plasmas, sometimescalled a vacuum spark, a plasma focus, etc Other more modern plasmas areproduced by irradiating a solid or gas or even clusters by short-pulsed laserradiation These plasmas occupy quite a large area of the parameters: the nature

of a plasma strongly depends on the experimental conditions, and its parametersare different during laser irradiation and in the decaying period after that Thesehigh-energy-density plasmas may be used as an x-ray light source or even as anx-ray laser source

Plasmas in nature should not be forgotten It is sometimes said that more than

99 percent of the material in the universe is in the form of plasma Just twoexamples are given Fig 1.2 The Earth is surrounded by several layers of iono-sphere It starts at about 100 km above the Earth's surface and extends up to some

500 km Another example is the solar corona surrounding the Sun; this plasmagreatly inspired the development of plasma spectroscopy

1.3 Nomenclature and basic constants

In this book the term plasma has dual meanings; the first is in the ordinary sense to

express a material which is composed of electrons, ions, and some neutral atoms

TABLE 1.1 Basic constants.

or even molecules Several examples are shown in Fig 1.2 The second usage is toexpress an ensemble of atoms or ions immersed in a plasma The latter may soundstrange, but it will turn out that this is rather natural

As we have already seen important parameters to characterize a plasma (in thefirst sense) are «e and T e Since electrons are usually much more active than ions

and neutrals in determining excited level populations, a plasma in the first sense

sometimes means simply an electron gas having a certain n e and T e

Constants which are used in this book without explanation are given inTable 1.1 The Rydberg constant, which is virtually equal to the ionization

potential of neutral hydrogen, is expressed as R Figure 1.8 shows the schematic

energy-level diagram of ions with several symbols Suppose we are interested in

the ion or the atom denoted with (z — 1) z indicates the charge ze of the ions in the

next ionization stage, or roughly speaking, the effective core charge felt by the

optical electron of the (z — 1) ion Here we call the electron that plays the inant role in a transition and emits radiation the optical electron If we are treating

dom-singly ionized helium in Figs 1.5 and 1.6, for example, (z — 1) is 1 (i.e dom-singly

ionized) and z is therefore 2 n z indicates the density of the ions in the nextionization stage For hydrogen-like ions (and neutral hydrogen), with which the

dominant part of this book is concerned, z is equal to the nuclear charge, p or q is used to indicate a discrete state n z _i(p), g z -i(p) and XZ-I(P) indicate, respectively,

the population (in units of m~3), the statistical weight and the ionization potential

(a positive quantity), of level p E z _i(p,q) is the energy difference between the

lower level p and the upper level q, i.e E z _i(p,q) = Xz-i(p) — Xz-i(<l)- The

sub-script z — 1 is omitted whenever confusion is unlikely to occur

1.4 z-scaling

For neutral hydrogen and hydrogen-like ions all the radiative transition

prob-abilities and the collision cross-sections scale according to z except for the

Boltzmann's constantdielectric constant of vacuumfine structure constantfirst Bohr radiusRydberg constant

NEUTRAL HYDROGEN AND HYDROGEN-LIKE IONS 15excitation and ionization cross-section values near the threshold; they haveappreciable z-dependences (Chapter 3) This latter point will be shown in Fig 3.11later As will be discussed in Appendixes 3A and 5B, these scaling properties lead

us to a certain scaling law for the plasma parameters

In Fig 1.2 an oblique line is drawn The position of this line does nothave any significance, but its slope and the scale attached on it are important.The slope is 2/7 on the logarithmic scales of this figure If we are interested in agroup of ions having a certain z, e.g z = 26 for hydrogen-like iron, present in a

plasma which have a certain T e , these ions feel and behave as if neutral hydrogen,

z= 1, feels and behaves in a plasma having re/z2 So, T e of the plasma in thefirst sense scales according to z2 Likewise, «e scales according to z7 Later in this

book, the quantities T e /z 2 and n e /z 7 are called the reduced electron temperatureand the reduced electron density, respectively The line in the figure shows this

scaling: hydrogen-like iron ions in a plasma with n e = 1025 m~3 (1019 cm~3) and

Jg = 108 K (9 keV) behave like neutral hydrogen would do in a plasma with

«e= 1.2 x 1015 irT3 (1.2 x 109 cm"3) and T e = 1.5 x 105 K (13 eV) On the basis ofthis scaling, we can infer the characteristics of the former ions from our knowledge

of the latter It is noted, however, that this simple scaling law is by no meansperfect This is partly because of the z-dependent cross-section values near thethreshold, as noted already Another more important exception is the ionization-recombination relationship, as will be seen in Chapter 5 Even so, this scaling law

is useful to understand, or estimate, qualitatively the properties of the ions underconsideration

In this section, we have been mainly concerned with neutral hydrogenand hydrogen-like ions Even for ions other than hydrogen-like, the properties

of excited states are not much different from those of hydrogen-like states,especially for highly excited states In this sense, the above scaling law is also validfor nonhydrogen-like ions to a certain extent For this reason many formulasderived in Chapters 4 and 5 are expressed in terms of the reduced temperature anddensity

1.5 Neutral hydrogen and hydrogen-like ions

We start with the classical picture: an electron with charge —e moves in the

Coulomb attraction field exerted by the ion with charge ze, fixed at the origin The

orbit of the electron is elliptic and it may be specified by the major axis, ellipticity,and the direction of the axis of revolution Instead of these parameters we choose

as parameters the total energy E, i.e the sum of the kinetic and potential energies, which is negative; the period of one revolution over the orbit r; the angular momentum A; and its projection onto a certain direction, say, the z-axis, /j, x We

take the quantity — 1Er as the ordinate in Fig 1.1 l(a) The abscissas are 2-TrA and 2-Kfix- Then all the possible orbits are included within the volume enclosed by three planes: two vertical planes that include the ordinate axis and the JJL\ = ±A

points; these planes correspond to the orbits with the axis directed parallel or

FIG 1.11 (Continued)

NEUTRAL HYDROGEN AND HYDROGEN-LIKE IONS 17antiparallel to the z-axis Another plane is the oblique plane for which 2-7rA =

—1Er holds; this plane corresponds to circular orbits All these parameters are measured in units of h We divide the whole volume into elementary volumes by using half-integer values of —2Er/h and 1-n \i\jh and integer values of 2ir\ Two

examples of the elementary volumes are shown in this figure We allocate an

integer n to the volumes of (n — 0.5) < (—2Er/h) <(n + 0.5) and name this integer

the principal quantum number The classical states with —2Er/h<Q.5 are not

allowed in reality We allocate an integer /to the volumes with /< (2-K\/ti) <(/+!) and call it the angular momentum quantum number The integer azimuthal quantum

number m t is allocated to (w/—0.5)<(2-7r/iA//i)<(w/ + 0.5) m t is also called the

magnetic quantum number The examples of the elementary volumes in this figure

are (n, I, mf) = (3,1,0) and (3,2,1) It is noted that all the elementary volumes, or the cells, have a volume of h 3 This cell may be regarded as a quantum cell; all the

states of electron motion expressed by points within a cell cannot be distinguished

in reality and therefore they are regarded as a single state Thus we have quantizedthe classical states, and we represent all the classical states within a single quantumcell with a point at the center of this cell Owing to the internal structure of theelectron, the spin, each point is doubly degenerate Suppose that the whole volume

is transparent except for these points If we look through Fig 1.11 (a) from the

direction of the /j, x axis and rescale the ordinate according to the energy E, we obtain Fig 1.1 l(b), the Grotorian diagram The levels having different m t belonging to a single / level are called the magnetic sublevels The state (n, /, mf) has the principal quantum number n, angular momentum Ih, and the projection of the angular momentum onto the quantization axis mfi In reality, the magnitude of the angular momentum is \/l(l + l)h.

FIG 1.11 (a) The quantization scheme of the classical electron motion of a

hydrogen-like ion The principal quantum number n is defined from the energy

(with the sign reversed) multiplied by the duration of one revolution of theelectron in the orbit; the angular momentum quantum number / is defined fromthe angular momentum A; and the azimuthal quantum number (the magneticquantum number) w/ is the projection of the angular momentum onto the

quantization axis, /JA- Each elementary cell has volume h 3 The two examples of

the quantum cells are shown for («,/, w/) = (3,1,0) and (3,2,1) Thequantization sheme is due to Professor R More, (b) The energy-level diagram

(the Grotorian diagram) of neutral hydrogen States with different m t in (a) arereduced to a single / level, but different / levels are resolved Different / levelscan be further reduced to a single level to lead to a simplified energy-leveldiagram like the one shown in Fig 1.6 Note that, in the case of neutralhydrogen, the energy is reduced to about one-quarter of that in Fig 1.6 In this

case, a level is specified with the principal quantum number n, or in later chapters, p or q This scheme is appropriate when we can assume statistical

populations among the different / levels in (b), or equal populations among the

different m t and / states belonging to the same n in (a).

See Table 1.1 for R, In terms of the classical Bohr picture, the radius of the circular

orbit (or the semi-major axis in the case of an elliptic orbit) is given by

It is noted that eq (1.4) is valid also for elliptic orbits The energy separationbetween the levels with principal quantum numbers differing by one is

In a plasma, electron, and ion collisions, which are random, tend to populate the

different / and mi states having the same n with equal probabilities In the classical

picture of Fig 1.11 (a), the populations are uniformly distributed among the same

n cells We say in this situation that the population is distributed statistically In

this case the total population of level n is 2«2 times the population in a single cell,where 2«2 is the number of the quantum cells, i.e n 2 times the degeneracy factor 2

If the statistical populations are actually established (its validity will be discussed

in Appendix 4A; see Fig 4A.2) the principal quantum number is enough to specify

a level In such a case p or q is understood to denote the principal quantum

* One atomic unit length is a 0 , and one atomic unit velocity is £ One atomic unit time is a 0 /£, or

r au = h/2R = 2A2 x 10~17s This corresponds to the time for the n= 1 electron to traverse one radian

length over the circular orbit of radius GO- Then the relationship corresponding to eq (1.6) becomes r™ • A_£"(«) = fi, where r™ is defined from r similarly to eq (1.4).

The energy of levels having different / and mi but the same n is given by

The speed of the electron in the circular orbit is

For a 0 and a see Table 1.1 The period of one revolution of the electron over the

orbit is

with

This quantity is also understood as an energy width allocated to the level having

principal quantum number n See Fig 1.11 (a) again It is interesting to find that

the two quantities of eqs (1.4) and (1.5) satisfy the relationship*

NON-HYDROGEN-LIKE IONS 19

number in place of n, so that, for example,/? = 1 means the ground state Therefore,

gz-i(p) = 2p 2 and XZ-I(P) = z 2 R/p 2 The Grotorian diagram of Fig 1.1 l(b) reduces

to a simplified energy-level diagram which is equal to Fig 1.6 except that the

ordinate is reduced to one quarter We adopt the convention that p < q means

hydrogen-are other examples For these atoms and ions an excited level (denoted by p) is designated by the principal quantum number n of the excited electron, the sum of the orbital angular momenta Lh* of all the electrons, and the sum of the spin angular momenta Sh* This scheme of combination of the angular momenta is called the L— S coupling, which describes well the energy-level structure of these atoms and ions A level is designated by n (2S+ l) L The suffix (25* + 1) is called the

multiplicity; S = 0 is a singlet, ^ is a doublet, 1 is a triplet, and so on To levels with

L = Q, 1, 2, 3, we assign the symbol S, P, D, F , , respectively.1" Figure 1.4

carries this nomenclature The level p= n (2S+l) L has the statistical weight g(p) = (2S+ 1)(2L + 1) This level is further split into the fine-structure levels, and

each component is designated by the total angular momentum Jh* In the case of

L>S, we have J=L — S, L S+l, ,L + S, These fine-structure levels are

designated by n (2S+r> L J For example, in Fig 1.4, the 23P ("two triplet P") levelconsists of three closely lying levels 23P0, 23P1; and 23P2 The statistical weight of

each of them is (2/+ 1), and their sum XX2/+ 1) is equal to (2S + 1)(2L+ 1) In the present example, g(2 3P) = 9 This designation is also adopted for hydrogenatoms and hydrogen-like ions Figure l.ll(b) carries this nomenclature In this

case, L = / and S = ^.

We defined the optical electron as the electron playing the dominant role inmaking a transition and emitting radiation We also define z; this quantity indic-

ates the effective core charge ze felt by the optical electron when it is at a large

distance from the core This happens to be equal to the roman numeral used todenote the ionization stage of a spectral line, e.g HI, CIII, OV

* As noted above, the actual magnitudes of these angular momenta are \/L(L + l)fi, \/S(S + l)h, and \/J(J + 1), respectively.

t These symbols stem from the early nomenclature of the series lines, "sharp", "principal",

"diffuse", and "fundamental" These characteristics can be recognized in Fig 1.3: in the wavelength region the series of pairs of sharp and diffuse lines are seen, each originating from the S and

long-D levels, respectively, in Fig 1.4 Other lines in Fig 1.3 are of the principal series, which originate from

P levels in Fig 1.4.

Here 6 (= n — «*) is called the quantum defect and n* the effective principal quantum

number Again for s states 8 is large and n* is appreciably smaller than n For d and

higher-/ states 8 is very small and n* is almost equal to n An example is seen in

Fig 1.4

In the following even in the case of nonhydrogen-like ions p = 1 is understood

to denote the ground state

As seen in Figs 1.4 and 1.1 l(b) high-lying levels form a series of levels verging to the ionization limit Their energies measured from the limit are given by

con-eq (1.1) or con-eq (1.7) (or con-eq (1.7a)) with large values of n We call these levels the

Rydberg levels (states).

In atomic spectroscopy an emission (absorption) line and the correspondingtransition is customarily written like: Hel A 318.8 nm (23S—43P); this means thatthis spectral line (one of the lines in Fig 1.3) is of neutral helium (called the firstspectrum) with wavelength 318.8 nm for transition with lower level 2 3S and upperlevel 4 3P The lower level comes first In the following we follow this convention.Figure 1.3 carries an alternative notation Instead of nm, units of A are sometimesused; 1 A is 0.1 nm

The first prominent line terminating on, or, in the case of absorption, starting

from, the ground state is called the resonance line An example is the Hell A 30.3 nm

(12S—22P) line shown in Fig 1.5 and identified in Fig 1.6 Another example is thetransition in Fig 1.4 of Hel (1 :S—2 :P) with the wavelength of 58.4nm

Finally, units of energy are mentioned 1 au (atomic units) is equal to

2_R = 27.2eV When R is used as units of energy (Rydberg units) energy is

expressed as Ry Units cm^1 is sometimes used to express a spectral line frequency

(v), and equal to vie in the cgs units 1 cm^1 is the energy difference corresponding

to a transition wavelength of A= 1 cm in vacuum; i.e 1 Ry= 1.0974 x 105cm^1

The energy of a level p with principal quantum number n may be expressed as a

modification of eq (1.1) by

where the parameter 0 is introduced which accounts for the degree of completeness

of the screening of the nuclear charge by the electrons other than the optical

electron If the screening is complete 0 equals zero For the optical electron having

an orbit penetrating deep into the core electron orbits, e.g the s electrons, (see

Fig 3.2(b) later), the screening is not complete, and 0 is a positive quantity, usually

smaller than 1 An alternative way of expressing the energy is

REFERENCES 21

References

Several books are available for plasma spectroscopy in general and for its variousfacets which are treated in the later part of this book A few of them are listedbelow

Cooper, J 1966 Rep Prog Phys 22, 35.

Griem, H.R 1964 Plasma Spectroscopy (McGraw-Hill, New York).

Griem, H.R 1997 Principles of Plasma Spectroscopy (Cambridge University,

Cambridge)

Huddlestone, R.H and Leonard, S.L (eds.) 1965 Plasma Diagnostic Techniques

(Academic Press, New York)

Lochte-Holtgreven,W.(ed.)l 968 Plasma Diagnostics (North-Holland, Amsterdam).

There are excellent books on atomic structure and atomic spectra Only a few arementioned

Bethe, H.A and Salpeter, E.E 1977 Quantum Mechanics of One- and Two-Electron

Atoms (Plenum, New York; reprint of 1957).

Condon, E.U and Shortley, G.H 1967 Theory of Atomic Spectra (Cambridge

University Press, London; reprint of 1935)

Hertzberg, G 1944 Atomic Spectra and Atomic Structure (Dover, New York) Shore, B.W and Menzel, D.H 1968 Principles of Atomic Spectra (John Wiley and

Sons, New York)

Thorne, A., Litzen, U and Johansson, S 1999 Spectrophysics (Springer, Berlin) White, H.E 1934 Introduction to Atomic Spectra (McGraw-Hill, New York).

THERMODYNAMIC EQUILIBRIUM

with normalization, ff(E)dE= 1 The average energy is E = 3kT e /2.

Boltzmann and Saha-Boltzmann distributions

In this book, the word "ions" is used to denote both ions and neutral atoms.However, the word "atoms" is used when it is more convenient to distinguishatoms and ions in adjacent ionization stages

It is well known from statistical mechanics that, in thermodynamic rium, the ratio of the number of ions per unit volume in two different energylevels, or the population density ratio, is given by the Boltzmann distribution

equilib-where we have assumed levels/? < q in ionization stage (z — 1) See Fig 1.8 Actual

level schemes are shown in Figs 1.4, 1.6, and l.ll(b) If the ions in this ionization

stage have many levels including p and q, we may plot the populations per unit

statistical weight in a semilogarithmic plot This plot is called the Boltzmann plot,

2.1 Velocity and population distributions

Maxwell distribution

In this book we consider a plasma, in the first sense, as consisting of electrons,

ions, atoms, and even molecules If n e is high and T e is low so that the meandistance between electrons becomes comparable to or shorter than the de Broglie

wavelength of electrons with thermal energy, A = h/'^fI^nnkT~ e , quantum effects

prevail, and the electron velocity distribution is given by the Fermi-Dirac tribution In the following we assume the opposite, i.e low «e and high T e :

dis-Then, we have the classical Maxwell-Boltzmann distribution (called simply theMaxwell distribution)

which satisfies the normalization condition, ff(v)dv = 1 The average speed

is v = ^/SkTe/irm, the root mean square speed is \/(v} 2 = ^/3kT e /m, and the

most probable speed is t>p = ^/2kT e /m The corresponding energy distribution

function is

VELOCITY AND POPULATION DISTRIBUTIONS 23

and we obtain a straight line, the slope of which corresponds to T e We will see

examples later in Chapters 4 and 5

Equation (2.3) applies to levels of ions in a particular ionization stage If we

take a hydrogen-like ion (z — 1), each of these levels corresponds to a level as

shown in Fig 1.6 The above thermodynamic relationship may be extended tohigher energies across the ionization limit to the continuum states of the electron,having positive energies We approximate here the continuum states as free-electron states Discussions concerning this approximation will be given inChapter 9 Since the level energy is continuous we consider free states of electrons

having speed v within the range dv This upper "level" is regarded as the collection

of states of free electrons paired to the core ion (in the ground state) in the

ionization stage z Then eq (2.3) is rewritten as

where n z (l,v)dv and g z (l,v)dv denote, respectively, the "population" and the

"statistical weight" of the upper "level", and A£" is the energy difference between

this upper level and the lower level p, i.e AE = mv /2 + x z

-i(p)-We now introduce the phase space, i.e a six-dimensional space for the motion

of a free electron: three dimensions for the spatial coordinate (x,y,z) and three dimensions for the momentum coordinate (p x ,p y ,p z ) In the x-p x plane, we define

a cell Sx • Sp x having area h This is another quantum cell (remember Fig l.ll(a))

and has the significance that all the states of motion, the corresponding points

(x, PX) of which fall in this cell, are regarded as a single state Similar arguments

apply to the y—p y and the z—p 2 planes Thus, the "number of states" deriving fromthe motion of the electrons is given as

where Ax, Aj, and Az are the spatial coordinate widths allocated to one of the

free electrons and t±p x , Ap y , and Ap z are similarly the momentum coordinate

widths The former widths make a volume A V allocated to the electron, which is equal to l/n e Since we assume the electron motion to be isotropic we use the polar

coordinate system

Equation (2.5) gives the number of states for the electron motion The "statisticalweight" of the "upper level" is thus given as

where g e and g z (T) are the statistical weights originating from the inner structure of

an electron and that of the ground-state ion, respectively The former comes from

the electron spin and is g e = 2, It is noted that eq (2.5a) is also encountered in solid

state physics as the density of states of electrons in the free-electron model.Equation (2.4) is transformed as

Equation (2.7) or (2.7a) is called the Saha-Boltzmann distribution Under certainconditions, thermodynamic equilibrium may be established in our plasma and the

population of an excited level p is actually given by eq (2.7) or (2.7a) If this is the case, we say that "level p is in local thermodynamic equilibrium (LTE) with

respect to ion z." This situation is called partial LTE In the case that the LTE

population, eq (2.7a), extends down to the ground state p=\, this situation is

defined as complete local thermodynamic equilibrium (complete LTE) Theseproblems will be treated later in Chapter 5

or

where we have used eq (2.2) In many situations we are not interested in the

"population ratio" as given by eq (2.6) Rather, the quantity of interest would be

the ratio of the "ion" density n z (l) and the "atom" density n z _i(p) in level p The

former quantity is obtained by integration of the "populations" over the speeds of

the free electrons, i.e n z (l) = fn z (l,v)dv By using the normalization condition for

the Maxwell distribution we obtain the Saha-Boltzmann distribution

or

where Z(p) is called the Saha-Boltzmann coefficient It is interesting to note that

Z(p) is expressed in terms of the thermal de Broglie wavelength (see eq (2.1)),

BLACK-BODY RADIATION 25

In traditional plasma spectroscopy, the term Saha equilibrium has been, and

still is, used to describe the density ratio of the "atoms" (z — 1) and the "ions" z,when complete LTE is assumed for this system The "atom density" is the sum ofall the "atomic" level populations,

The summation in the r.h.s (right-hand side) of this equation is called the partition

function, and denoted as B z _i(T & ) We define the "ion" density N z in a similar

manner by introducing the partition function of the ions B z (T e ), Then, the density

ratio of the "atoms" and the "ions" is given as

It is readily seen that, because of the nature of the statistical weight, g(p) = 2p 2

for the case of hydrogen atoms and hydrogen-like ions, the partition functionsdiverge This difficulty comes partly from eq (2.8) itself, i.e the notion that all thepopulations in excited levels of atoms belong to the "atom." We will see inChapters 4 and 5 that this understanding is rather unrealistic; excited atoms arestrongly coupled to ions rather than to the ground-state atoms The difficulty ofthe divergence itself will be resolved toward the end of Chapter 9

One important fact is noted here: in deriving eq (2.6) we "filled" the density

of states for free electrons, eq (2.5a), with the Boltzmann distribution, eq (2.4),with appropriate statistical weights incorporated Equation (2.6) itself includesthe Maxwell distribution function, eq (2.2) This fact clearly indicates that theMaxwell distribution is nothing but the Boltzmann distribution extended overthe free-electron states The normalization factor in eq (2.2) or eq (2.2a) makesthis point less obvious This aspect will be further examined in Chapter 9 See

eq (9.19a)

2.2 Black-body radiation

We consider an ensemble of ions having lower level p and upper level q, and the

radiation field, the wavelength of which corresponds to the transition energy

between these levels The temporal development of the upper-level population n(q)

is given by the rate equation

where /„ [Wm 2sr :s] (sr means steradian or a unit solid angle) is the spectral

intensity of the radiation field at the transition frequency v = E(p, q)/h Figure 2.1 illustrates the situation of eq (2.10) The quantity I v is called the spectral radiance

FIG 2.1 The emission-absorption processes of atoms in a radiation field.

in radiometry It should be noted that we have assumed that, in eq (2.10), theradiation field is isotropic and has virtually a constant intensity over the lineprofile of the transition.* The first term represents excitation of the upper-levelions by absorption of photons, and the second and third terms denote deexcitation

by spontaneous transition and by induced emission, respectively A(q,p), B(p, q) and B(q,p) are called Einstein's A and B coefficients We further suppose that this

system is surrounded by "mirror" walls that reflect radiation and ions completely.After a sufficiently long time a stationary state is reached and the time derivativevanishes Then we have

On our above assumptions, we may expect that our system is in dynamic equilibrium Then, the population ratio should be given by the Boltzmanndistribution, eq (2.3),

thermo-* Equation (2.10) cannot describe the atomic system in a radiation field that violates these conditions This is the case for radiation fields encountered in many practical cases; an extreme example is the field produced by a laser beam This field is highly directional, monochromatic, and sometimes polarized In such cases we have to employ an alternative approach by introducing the concept of an absorption cross-section for the transition, as defined by eq (3.9) later.

t Equation (2.10) is sometimes written in terms of the spectral energy density, (4w/c)I v , in place of the spectral intensity I Then the second relationship of eq (2.12) takes a different form.

We note the interrelationships between the coefficients A and B*

This distribution is called the Rayleigh-Jeans law, and is of an entirely classical

nature as is understood from the absence of h This distribution diverges with v.

At the other extremum, for hv ;$> kT, we obtain

Equation (2.18) is called the Stefan-Boltzmann law

region corresponding to the transition p <-> q We may readily extend the above

argument to other regions by introducing other transition frequencies, and finally,

to obtain eq (2.14) for the whole spectral range Equation (2.14) is called Planck'sdistribution or the black-body radiation Figure 2.2(a) shows examples of theblack-body radiation for several temperatures

Several properties of the black-body radiation are discussed It has a maximumintensity at a certain frequency, depending on the temperature The frequencywhich gives the maximum is readily obtained from the derivative of eq (2.14),

See Fig 2.2(a)

In the low-frequency region oihv^_kT, eq (2.14) reduces to

The energy density of the black-body radiation contained in a unit volume is

(^/c)B v (T)dv, in units of [Jm~3], and the total energy density is